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Existence of \(\partial\)-parameterized Picard-Vessiot extensions over fields with algebraically closed constants. (English) Zbl 1280.12003

The paper under review is devoted to the proof of the existence, in the case of one differential parameter \(\partial\), of \(\partial\)-parameterized Picard-Vessiot extensions for systems of linear difference-differential equations over difference-differential fields with an algebraically closed field of constants. The main result is as follows. Let \(K\) be a field of zero characteristic considered together with sets of its automorphisms and derivations, denoted by \(\Sigma\) and \(\Delta\), respectively, and with a derivation \(\partial: K\rightarrow K\) such that any two operators of the set \(\Sigma\cup\Delta\cup\{\partial\}\) commute (such a field is called a \(\Sigma\Delta\partial\)-field). Then for every \(\Sigma\Delta\)-linear system over \(K\), there exists a \(\partial\)-parameterized Picard-Vessiot ring \(R\) such that the ring of \(\Sigma\Delta\)-constants of \(R\) is a finite algebraic field extension of the field of \(\Sigma\Delta\)-constants of \(K\).
In this formulation, a \(\partial\)-parameterized Picard-Vessiot ring for a \(\Sigma\Delta\)-linear system \(\sigma_{i}(Y) = A_{i}Y\) (\(A_{i}\in\mathrm{GL}_n(K)\), \(\sigma_{i}\in \Sigma\)), \(\delta_{i}(Y) =B_{i}Y\) (\(B_{i}\in K^{n\times n}\), \(\delta_{i}\in \Delta\)) with integrability conditions \(\sigma_{i}(A_{j}) =\sigma_{j}(A_{i})A_{j}\), \(\sigma_{i}(B_{j})A_{i} = \delta_{j}(A_{i}) + A_{i}B_{j}\), \(\delta_{i}(A_{j}) + A_{j}A_{i} = \delta_{j}(A_{i}) + A_{i}A_{j}\) (\(\sigma_{i},\, \sigma_{j}\in\Sigma\), \(\delta_{i},\,\delta_{j}\in\Delta\)) is defined as a \(K\)-\(\Sigma\Delta\partial\)-algebra \(R\) such that
(i) \(R\) is \(\partial\)-generated by a fundamental solution matrix for the system, that is, there exists \(Z\in \mathrm{GL}_n(R)\) such that \(\sigma_{i}(Z) = A_{i}Z\), \(\delta_{i}(Z) = B_{i}Z\) (\(\sigma_{i}\in\Sigma\), \(\delta_{i}\in\Delta\)) and \(R = K\{Z,\frac{1}{\det(Z)}\}_{\partial}\) and
(ii) \(R\) is \(\Sigma\Delta\)-simple.
This definition slightly deviates from the corresponding definition of C. Hardouin and M. F. Singer [Math. Ann. 342, No. 2, 333–377 (2008; Zbl 1163.12002)] and P. J. Cassidy and M. F. Singer [IRMA Lect. Math. Theor. Phys.9, 113–155 (2007; Zbl 1230.12003)] where (ii) is replaced with the condition that \(R\) is \(\Sigma\Delta\partial\)-simple. Note that the above mentioned theorem of the paper under review leads to an essential improvement of the results of these two works in which the Galois theory of parameterized differential equations is built under the assumption that the filed of constants is differentially closed.

MSC:

12H05 Differential algebra
12H10 Difference algebra
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References:

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